- Commit
- 65e93388cd80c096674dbfa5510dedb2ce0c8165
- Parent
- c37732113c53393e5b48794104a7547dccf0cfc0
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Added some qualifiers to the statement about the Lie functor for algebraic groups
lie-algebras-and-their-representations
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Diffstat
1 file changed, 4 insertions, 3 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/introduction.tex | 7 | 4 | 3 |
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -274,9 +274,10 @@ there is no rational group homomorphism \(\mathbb{C} \to In particular, the Lie functor \(\mathbb{C}\text{-}\mathbf{Grp}_{\operatorname{simpl}} \to \mathbb{C}\text{-}\mathbf{LieAlg}\) -- between the category -\(\mathbb{C}\text{-}\mathbf{Grp}_{\operatorname{simpl}}\) of complex algebraic -groups and the category of complex Lie algebras -- fails to be full. Similarly, -the functor \(\mathbb{C}\text{-}\mathbf{Grp}_{\operatorname{simpl}} \to +\(\mathbb{C}\text{-}\mathbf{Grp}_{\operatorname{simpl}}\) of simply connected +complex algebraic groups and the category of complex Lie algebras -- fails to +be full. Similarly, the functor +\(\mathbb{C}\text{-}\mathbf{Grp}_{\operatorname{simpl}} \to \mathbb{C}\text{-}\mathbf{LieAlg}\) is \emph{not} essentially surjective onto the subcategory of finite-dimensional algebras: every finite-dimensional complex Lie algebra is isomorphic to the Lie algebra of a unique simply